A333054 Numbers m such that r(m) > r(k) for all k < m, where r(m) = min(sigma(m)/m, sigma(m+1)/(m+1)), and sigma(m) is the sum of divisors of m (A000203).

1, 2, 8, 14, 44, 104, 495, 944, 4095, 5775, 5984, 21735, 98175, 862784, 4096575, 7194824, 14753024, 879207615, 1969789184, 2275962975, 3968862975, 12567844575, 39566665215, 44803620225, 77510285775, 125617830975, 162902829375

Offset: 1

Author

Amiram Eldar, Mar 06 2020

Keywords

• nonn
• more

Comments

The corresponding values of r(a(n)) are 1, 1.333..., 1.444..., 1.6, 1.733..., 1.828..., 1.890..., 1.970..., 1.999..., 2.044..., 2.085..., 2.120..., 2.181..., 2.243..., 2.248..., 2.252..., 2.360..., 2.397..., 2.407..., 2.408..., 2.411...

The least number m such that both m and m+1 are k-abundant (i.e., their abundancy indices sigma(m)/m > k and sigma(m+1)/(m+1) > k) is a term in this sequence. E.g., a(10) = 5775 = A096399(1).

a(28) > 5*10^11. - Amiram Eldar, Jan 02 2021

Examples

			The values of min(sigma(k)/k, sigma(k+1)/(k+1)) for k = 1, 2, ... 8 are 1, 4/3, 4/3, 6/5, 6/5, 8/7, 8/7, 13/9. The record values in this range, 1, 4/3 and 13/9, are obtained at k = 1, 2, and 8.
		

Crossrefs

Cf. A000203, A004394, A068403, A096399, A333053.

Programs

Mathematica

seq={}; rminmax = 0; r1 = 1; Do[r2 = DivisorSigma[1, n]/n; rmin = Min[r1, r2]; If[rmin > rminmax, rminmax = rmin; AppendTo[seq, n-1]]; r1 = r2, {n, 2, 10^6}]; seq

Extensions

a(22)-a(27) from Amiram Eldar, Jan 02 2021